The Infinite, part 2. Parsing infinity – In the Beginning.

In the last post, we looked a little at meanings. What do we mean by “finite”? And the answer was that it depends. If we are measuring size, it is a matter of counting: counting is just a matching exercise. Match numbers to the number of cows that pass the gate for example: one, two, three, . . . 25. 25 cows came through the gate. Our ordinary experience prepares us for such things. But when the number of objects becomes too large, the process becomes less meaningful. Scriptural accounts that suggest certain things are just too large to comprehend can be understood on several levels. Whether they entail the infinite will be examined later. Questions like “How many moons does Jupiter have?” and “How many water molecules are in a cup of water?” are not just different in scope, they are different in meaning. Abstraction and approximation are the only ways to deal with the second question. (The “answer” is *about* 8 x 1024. Ten to the 24th power is so large that we can only deal with it as an abstraction. But it is a finite number!)

Some cultures avoid counting things when they are too large in size. But the accountants won’t give us that luxury now. Budget and deficit and loss discussions bat around extraordinary figures. Our common experience does not prepare us to understand the idea of a trillion dollars and it may be impossible to do so. So we deal with these kinds of things as abstractions. Does that make you a bit nervous?
In part 1, without being too explicit about it, we really separated the concept of the infinite into two categories. We might call them the qualitative infinite and the quantitative infinite. Qualitative infinity is reserved for later. Size is more easily defined so I will consider a few ideas about the quantitative side first and I will continue that in the next two posts as well.

Can, or *should* one consider collections of things which are infinite in size? We know from part 1 what it must mean for a collection of things to be infinite in size.
An example would be the collection of counting numbers: {1, 2, 3, . . . }.[1] In this case the “. . .” is a comforting euphemism for “everything else after 1, 2, 3”! The idea that one should try to deal with non-finite collections of things is fraught with logical peril. But we shall be brave, or stupid, depending on how your philosophy has been or will be shaped by the infinite. The difficulty begins to rear its head when we try to push our understanding of size for finite collections or sets of things, to infinite sets or collections.

How big *is* {1, 2, 3, . . .}? We can just say infinite, and drop it at that. But measuring finite sets was a correspondence game. Counting the cows involved simply matching numbers with cows as they come through the gate:

1 <-> cow1
2 <-> cow2
.
.
.
25 <-> cow25

The problem with {1, 2, 3, . . . } appears when we try to count it:

1 <-> 1
2 <-> 2
3 <-> 3
.
.
.

We never run out of things to count. On the other hand, does everything in {1, 2, 3, . . .} get counted by this matching? The answer is clearly, YES. You can’t name a number that fails to be matched in this scheme. We apparently had to use the whole of {1, 2, 3, . . .} to get a complete match (with itself).

So that I don’t have to keep writing {1, 2, 3, . . .} I will just use the symbol N for it. Saves ink.

Next, observe that we can match the whole of N with other things. Such as the numbers that are multiples of 5: {5, 10, 15, 20, 25, . . .} Consider:

1 <-> 5
2 <-> 10
3 <-> 15
4 <-> 20
5 <-> 25

etc.

This matching rule is obvious, and everything gets matched. 1024 for example gets matched with 5 x 1024. Think what this means: the multiples of 5 form an infinite collection, *exactly the same size as N.* But of course, the collection of numbers which are multiples of 5 is completely and *properly* contained in N. There is clearly more stuff in N. That’s odd. But for things of infinite size, the principle of measure seems to work a bit differently than it does for sets of finite size. Things that appear to be larger in one sense, may fail to be so in the “counting” sense.[2] Next up: really infinite stuff.😉

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[1] It has been argued that we should regard the counting numbers as only “potentially” infinite. You just make larger numbers when you need them. The platonists don’t care for this. This “finitism” may be satisfactory for some purposes, but may be inadequate to discuss Mormon theology for example.

[2] The size of N, measured by itself(!) is usually designated by the Hebrew letter alef, with subscript 0: אo. Observe then that the size of {5, 10, 15, 20 . . .} is also אo. There are אo things in N.